397
B+L Violation at High Energy
Larry McLerran Theoretical Physics Institute School of Physics and Astronomy University of Minnesota Church St. 116 Minneapolis, Mn . 55455
Abstract
Recent work which suggests that baryon plus lepton number might be violated in high energy scattering is discussed. The status of B+L violation at high temperatures is reviewed. 398
1 Introduction
In this talk, I will discuss the following issues:
1. Baryon number violation at finite temperature,and the reliability of computations of the rate. 2. The breakdown of perturbation theory in electroweak theory for B+L violating processes, and the possibility that B+L violating cross sections might be large in the collision of high energy particles.
I will begin by briefly reviewing the fact that B+ L is not conserved in electroweak theory due to an anomaly in the conservation of the baryon number current. Although the rate for this process is small at low energy and zero temperature, I will review considerations which argue that the rate is large for high temperature. I discuss some apparent paradoxes which arise from having a large rate at high temperature. The resolution of this paradox requires that there be large matrix elements for processes which involve a large number of particles. The implication of the finitetemperature result is that there might be processes which are large at high energy corresponding to multi-particle production amplitudes for B+L violation. This implication is discussed, as are recent instanton estimates of high such amplitudes. I argue that perturbation theory breaks down in this sector of the theory. A supersymmetric Yang-Mills example is presented where the rate for B+L changing processes can be shown to be large.
Baryon Number is Not Conserved in Electroweak Theory 2 The equations of motion for electroweak theory naively conserve baryon number. This should be true since we would like the rate to be very small, and since the scale of electroweak physics is a mere 100 GeV, and if there was explicit violation of baryon number, it should be observably large. When the equations of motion of electroweak theory are quantized, baryon number is no longer conserved. This is a consequence of the Adler-Bardeen anomaly.!1J-[3J. Baryon number is violated as 2 (1) 8,,J� = 3�7r2tr F" " F:v = 8,,K65 In this equation, F"" is the field strength of the electroweak W and Z fields, pd pM (2) µ,v = 2Eµ,v.\a1 399
The product F pd can be written as the divergence of the Chern-Simons current where I
fj.NB fj.Qcs dt d3x (5) = = j 8µI In Fig. 2b, we have deformed the W and Z fields from a minimum to a maximum separating two minima. In the process of this variation, the fermi levels move. As shown by Mantonf5] (for not too strong a scalar coupling [6]) the configuration which sits at the top of the barrier corresponds to a level crossing,\ of a fermion at zero energy. The configuration of W, Z and Higgs fields at the top of the barrier between the two minima is called the sphaleron. It was firstconstructed for electroweak theory by Klinkhammer and Manton.171 The sphaleron is a solution of the static classical equations of motion. Since it corresponds to a top of an energy barrier, it is classically unstable under time dependent perturbations. By making small fluctuations aroundthe sphaleron, it is also possible to construct the shape of the potential in the neighborhood of the top of the barrier. The sphaleron has Chern-Simons charge of 1/2. In Fig 2c, the W, Z and Higgs fields have been deformed to the minimum to the right of the original minimum. The fields arenow gauge transforms of the fieldsnear the original minimum, and therefore the spectrum of fermi levels is the same. The only difference is that in the deformation, a filled energy level which originally had negative energy has gained positive energy. A particle has been produced. This adiabatic level crossing is a general feature of electroweak theory. The relation between it and the original computation of the anomaly is not obvious. The original computation required a careful use of ultraviolet regulators. In this level crossing picture, it is necessary to know that such a regulation scheme exists, since we have ignored what is happening at the bottom of the fermi sea. The non-conservation of baryon number is related to transitions between degenerate minima of the effective potential with differing topological charge. At zero temperature, the only way such transitions can take place is by quantum mechanical tunneling, as shown in Fig. 3. This tunneling process was first computed by t' HooftlSJ. In the computation of the rate, a classical solution of the equations of motion is used which is called an instanton,19] which corresponds to the classical solutions used in the WKB method of ordinary quantum mechanics. The process computed by t' Hooft is therefore often referred to as instanton mediated. The result of t' Hooft 's computation is that the rate is (6) In this formula, there are no dimensional factors. Remember that even if we measure the space time volume of the universe in terms of the W boson compton wavelength, it is only 10168 (7) Vtuniverse """-' 401 Since there are in fact very few protons per Compton volume of the W boson, the rate given by t' Hooft is so small that never in the history of universe was it even marginally likely that an instanton generated a baryon number change. Nevertheless, the computation does prove that the rate of baryon number change in electroweak theory is non-zero. The process generated by instantons is visualized in Fig. 4. This amplitude violates both baryon number and lepton number by 3 units. The sum of baryon number plus lepton number is not conserved, but the difference B + L is conserved. This is generally true in electroweak theory. There is a corresponding anomaly for the lepton current which is identical with that of the baryon current. Note also, that each generation of lepton and baryon is produced in the process. The electromagnetic and color charges are also conserved. Even if such a process as shown in Fig. 4 existed at a large rate in electroweak theory, it would be very difficult to detect experimentally. Most proton decay experiments are designed to measure only single proton decays. Requiring a three body decay in a nucleus would surely reduce the rate due to the small probability that three nucleons overlap. In addition, requiring that only light mass quarks are produced would involve strong suppression due to weak mixing angles. Finally, there would in general be much missing energy in the event due to the large probability of emitting a neutrino. 3 The Rate of Baryon Number Non-conservation at Finite Te mperature At finitetemperature, an entirely new method of generating baryon number change arises. There is some probability that there is a thermal fluctuation with an energy higher than the height of the barrier between two degenerate minima. In this case, the fluctuation can classically traverse the maximum and enter the new minimum, as shown in Fig. 5. The probability that the thermal fluctuation has an energy higher than the top of the barrier is[7J,[10]-[ll] (8) As discussed above, if we know the sphaleron solution of the classical equations of mo tion with topological charge Q ph 1/2 6.Qc., then we know the height of the barrier s = Esph· The energy of the sphaleron comes from a classical solution, and classical solutions correspond to the limit of very many quanta aw, so that the energy is of order (N) � 1/ M Esph � w/aw � 10 (9) 1- TeV 402 The typical energy per quanta is of order Mw, and thereforethe size of the sphaleron is of order Rsph l/Mw (10) � We can visualize the process of making a classical transition over the top of the barrier by the diagram of Fig. 6. The diagram is identical to that of the instanton process, and the conservation laws associated with the event are identical. The sphaleron process may be thought of as analogous to an instanton process, except that we imagine the sphaleron process taking place in real time at finitetemperature. The above estimate of the probability that there is a thermal fluctuation with an energy larger than the height of the barrier clearly gives an overestimate of the rate. We must also know the probability that the thermal fluctuation is in some sense near the top of the barrier. For example, if we are lost in a forest in the high Himalayas, we may be much higher than a mountain pass in the Rockies, but we may never find our way to it. Put another way, the function space in which we imagine our thermal fluctuation is multi-dimensional and complicated. It is not obvious what weight a thermal fluctuation has for inducing a baryon number changing transition. There is a reliable method for computing thermal transition processes if the coupling is weak 12J-[13J. We must firstask if there is a range of temperatures where weak coupling ) methods are reliable. First we must recall that at high temperature, the mass of the W boson becomes temperature dependent Mw Mw (T).[141 At some temperature, the -+ masses of the W and Z bosons approach zero, and the broken symmetry of electroweak theory is restored. In general, it is possible to show that perturbation theory breaks down in electroweak theory when T Mw (T)/owJ15J-[l6J This is because infrared integrations over thermal propagators become:2: so singular that the factors of alpha generated at each order of the loop expansion are canceled. In addition, the use of the Langer-Affieck formalism for computing thermal transitions over the barrier is only valid in the high temperature limit, T Mw(T). This latter case is not much of a restriction since only in this limit are thermal>> transitions larger than quantum tunneling, that is e-Mw/T e-4,,./aw. We therefore conclude that there is a region of temperatures where the rate:2: of baryon number changing transitions may be reliably computed in electroweak theory, Mw (T) << T << Mw(T)/a.w (11) In the above range of temperatures, the rate of baryon number change has been com puted in electroweak theory for >../g 2 1 where >.. is the four Higgs coupling � itrength l 7]-[18] The rate is large compared to the expansion rate of the universe in ) 403 cosmology, r ;r 109 for T 100 Ge v (12) llB universe :'.'.". � The rate has also been recently computed for arbitrary values of with the result >..f g , that there is some suppression for values not close to one, but with essentially2 the same conclusions as was the case for >../g2 1 [l9]-[20J. � For temperature T :'.'.". Mw (T)/aw, it is not possible to do reliable weak coupling estimates. It is however possible to make estimates based on scaling arguments or ap proximate classical solutions)l 7],[21] The rate here is also many orders of magnitude larger than the expansion rate of the universe. The Problem with Instantons 4 The sphaleron estimates of the rate of baryon number change have been criticized by several authors.l22J-[23] The paper by Ellis et. al. points out several puzzling apparent paradoxes between instanton computations and those of sphalerons, and we will address these apparent puzzles here. The resolution of these apparent paradoxes sheds some interesting light on the mechanism of sphaleron transitions)l 7J. We will not discuss the second set of criticisms due to Cohen et. al., as this would go beyond the scope of this talk, except to say that this paper has, in my opinion, at least two essential mistakes which invalidate the conclusions, as has been recently pointed out in the work by Dine et. al., and by Mottola et. aJ.[24]-[25] The first set of apparent problems with the sphaleron computation is that the sphaleron is a large classical object. It has (N) � l/aw quanta in it and is large compared to its Compton wavelength. It should be hard to make such a fat, fluffy object. In fact, when the method of Langer-Affieck is applied to the computation of the rate of sphaleron processes, one estimates the probability of this production. There is a suppression, roughly proportional to al\,, which results. There are some large numerical factors however which enhance the rate. When all the factors are combined together, the overall suppression is about 10-5. This suppression is not enough to make the rate irrelevant cosmologically since the rate of expansion of the universe is slow compared to the natural time scale set by thermal processes T / Mplanck � 10-16. It is easy to get confused by this issue nevertheless. If we think in terms of exclusive probabilities, the probability to have only a sphaleron in a box the size of a sphaleron compared to that to have ordinary thermal fluctuations is very small P The � e-T3/Mi question we are asking is however an inclusive one. We ask what is the probability that 404 there is a sphaleron transition within the volume of the sphaleron plus all thermal excita tions within the same volume. The sphaleron is weakly interacting, and the probability that a highly excited Z or W boson interacts with it during its lifetime is small, that is, the sphaleron is transparent to the high energy modes and one can compute using inclusive probabilities. This probability is not a priori small)24J,[26J A more difficultinstanton argument to evade is provided by the following "theorem". Consider the baryon number changing expectation value where is a quark field (qqql) q and is a lepton field. Wewill consider for purposes of this argument a theory with only l one generation of quarks and leptons. We will consider this expectation value at finite temperature. This expectation value gives the amplitude for a baryon number changing process. We can estimate this expectation value using the path integral, j qqq/e-S (13) ( qqq/ ) - - j e-S This amplitude may be computed in Euclidean space, and later analytically continued to Minkowski space. The numerator of the above ratio of path integrals is nonzero only if (14) Now we also have (15) and the Higgs part of the action is positive definite. Therefore (16) and -2-rr/ow (17) (qqql) '.::'. e The conclusion seems to be that baryon number processes are exponentially suppressed in contradiction with the sphaleron arguments. How can this argument be wrong? One possibility is that instanton computations may break down in a weak coupling expansion for E Mw /aw, and that the estimate above is simply incorrect. 2 Even if the above argument is correct, it does not prove that all baryon number changing processes are suppressed.r26J Recall that the sphaleron is composed of many quanta. We would expect at high temperature that the dominant matrix elements involve of order l/aw quanta. If these matrix elements are Poisson distributed in the number of participating quanta (18) 405 so that while the total probability is of order one, the probability of a few prong event is exponentially suppressed (19) If the above situation is true, then it must also be true that if instanton methods are used to compute these multi-particle processes, the weak coupling expansion around the . instanton will break down, since instanton processes always have a prefactor of e-h/cxw. As we will see in a later section, this is what seems to happen when instantons are employed to estimate the rate of baryon number changing processes.127J-129J In these calculations, instanton methods are used to estimate matrix elements for processes which have baryon number change in the presence of a large number of quanta. If taken at face value, these calculations would imply that the rate is large. These calculations are however plagued by the breakdown of weak coupling methods, and conclusions are still in dispute. We can also see in an explicit quantum mechanics example that a large suppression as sociated with instantons does not imply a similar suppression for sphaleron processes.126J Consider the pendulum shown in Fig. 7. In a gravitational field, the potential for the pendulum is periodic as is shown in Fig. 8. The instantons of this theory are classical solutions which in Euclidean space interpolate between 0 0 and 0 2mr in a time 1/T. = = The sphaleron is the classical solution which sits on top of the barrier, 0 2(n + 1/2)7r. = We can easily compute both the sphaleron and instanton contributions at high tem perature, where the potential can be ignored. The sphaleron contribution is 1. e-E,p/T � The instanton classical solution is 0 2IrtT (20) = The Euclidean action is 1 S dt-0 27r (21) · � � 1/30 2 2 2 T Therefore at high temperatures, the instanton contribution is greatly suppressed, e-2"2T In the case of the simple pendulum at high temperature, we of course know the correct answer. The highly excited pendulum will roll over many times. The sphaleron therefore predicts qualitatively correct results where the instanton naively gives incorrect intuition. Instantons and High Energy Scattering 5 In Fig. 9, the amplitude for a baryon number changing process in the presence of a large number of scalars and gauge bosons is shown. Let us for definiteness assume that the number of scalars and gauge bosons is N. By crossing symmetry, this amplitude 406 describes a large number of particles going to a large number of particles, and as well two particles going to a large number. The difference between these amplitudes is that in the first case, we can have a transition where all of the particles have a low energy, E Mw, and where all invariant subenergies are small. In the latter case, the invariant � subenergies of all particles connected by interactions with one of the particles in the initial state has a large invariant subenergy. By the considerations of the previous sections, the amplitude for a large number of particles to go into a large number of particles with baryon number change must become large if the total energy is of order E Ew This does not necessarily imply that the � amplitude for a few particles to go into a large number is large. Because of the large invariant subenergies, the amplitude might be cut off by a variety of form factor like effects. In addition, at fixed energy, the phase space available per particle in the final state is less by about a factor of two, and since there are a large number of produced particles, the magnitude of the relative suppression(l/2)2N can be big. It is interesting that when the amplitude for few particle collisions to make large numbers of particles plus baryon and lepton number change are computed, one finds that naively the total cross section becomes large.l27J-[29J. This can be seen to be a consequence of the pointlike nature of instanton induced amplitudes. To leading order in the classical computation, the instanton induced amplitude for multiple scalar emission, ignoring the contributions arising from fermion zero modes, is [d l e -2"/e and again the cross section becomes of order l_[30J. For the realistic electroweak case however as yet uncomputable corrections to the formula for the total cross section make definite conclusions hard to establish, and there appear to be large corrections due to form factors.131]-[32]_ While it is not yet established that the cross section for baryon number violating processes do in fact become large in electroweak theory, it is amusing to speculate on the phenomenological consequences if such behavior does happen. We expect that unitarity breaks down at some scale of order E Mw /aw. In this case, the weak interactions � become strong. The total cross section is expected to be (1 / E P a?,v/ M'tv , a cross O' � � )2 � section typical of off resonance weak interaction processes. In baryon number changing processes, the total multiplicity of jets is large, N 1/aw, and the typical transverse � momenta of jets is expected to be Mw . The events would be quite spectacular, and common at high energy. Pt � 6 Acknowledgments I gratefully acknowledge my colleagues at Minnesota, L. Carson, Xu Li, A. Vainshtein, M. Voloshin, R. Wang with whom much of the work discussed here was carried out. I also grateful acknowledge many useful conversations with V. Zhakharov on the breakdown of perturbation theory around instantons. References [1] S. Adler, Rev. 177, 2426 (1969). Phys. [2] J. S. Bell and R. Jackiw, Nuovo Cim ento 51, 47 (1969). [3] W. Bardeen, Rev. 184, 1841 (1969). Phys. [4] J. Ambjorn, J. Greensite, and C. Peterson Nuc. B221, 381 (1983). Phys. [5] N. Manton, Rev. D28, 2019 (1983). Phys. [6] L. Yaffe, University of Washington Preprint (1989). [7] F. Klinkhammer and N. Manton, Rev. D30, 2212 (1984). Phys. [8] G. 't Hooft, Rev. Lett. 37, 8 (1976). Phys. [9] A. Belavin et. al., Lett. 59B, 85 (1975). Phys. 408 [10] S. Dimopoulos and L. Susskind, Phys. Rev. Dl8, 4500 (1975). [11] V. Kuzmin, V. Rubakov and M. Shaposhnikov, Phys. Lett. 155B, 36 (1985). [12] J. Langer, Ann. Phys. 41, 108 (1967); 54, 258 (1969). [13] I. Affleck,Phys. Rev. Lett. 46, 388 (1981). [14] D. A. Kirzhnits and A. Linde, Ann. Phys. 101, 195 (1976). [15] A. D. Linde, Phys. Lett. 96B, 289 (1980). [16] D. Gross, R. Pisarski and L. Yaffe, Rev. Mod. Phys. 53, 43 (1981). [17] P. Arnold and L. McLerran, Phys. Rev. D36, 581 (1987). [18] A. Ringwald, Phys. Lett. 201B, 510 (1988). [19] L. Carson and L. McLerran, Phys. Rev. D41, 647 (1990) [20] L. Carson, Xu Li, L. McLerran and R. T. Wang, University of Minnesota Preprint TPI-MINN-90/13-T [21) S. Khlebnikov and M. Shaposhnikov, Nuc. Phys. bf B308, 885 (1988) [22) J. Ellis, R. Flores, S. Rudaz, and D. Seckel, Phys. Lett. 194B, 241 (1987). [23) A. Cohen, S. Dugan and A. Manohar, Phys. Lett. 222B, 91 (1989). [24) M. Dine, Lechtenfeld, B. Sakita, W. Fischler and Polchinski, City College Preprint CCNY-HEP-89/0. 18 J. [25) J. Cline, E. Mottola and S. Raby, in preparation. [26) P. Arnold and L. McLerran, Phys. Rev. D37, 1020 (1988). [27) A. Ringwald, Nuc. Phys. B330, (1990). 1 [28) Espinosa, California Institute of Technology preprint CALT-68-1586. 0. [29) L. McLerran, A. Vainshtein and M. Voloshin, University of Minnesota Preprint TPI MINN-89 /36-T [30) L. McLerran, A. Vainshtein and M. Voloshin, University of Minnesota Preprint TPI MINN-90/8-T [31) V. Zakharov, University of Minnesota Preprint TPI-MINN-90/7. 409 [32] P. Arnold and M. Mattis, Los Alamos National Lab Preprint. Figures: Fig. 1 The minimal energy, or effective potential, as a function of 6.Qcs- The effective potential is multiply periodic. Fig. 2a The effective potential and the fermi levels for W, Z and Higgs fields near the minimum with zero topological charge. 2b The W, Z and Higgs fields have been deformed to near the top of the energy barrier. The fermi levels move, and an occupied state crosses zero energy. 2c The W, Z, and Higgs fields are now near the next minima and the topological charge has changed by one unit. An occupied level has crossed zero energy and is now at the position of the firstunoccupied level near the previous minimum. Fig. 3 Tunneling through the effective potential from one minimum to another. Fig. 4 The amplitude for baryon number change generated by instantons. Fig. 5 Transitions across the maximum between two degenerate minima of the effec tive potential for a thermal fluctuation. Fig. 6 A sphaleron induced baryon number violating process. Fig. 7 A simple pendulum. Fig. 8 The multiply periodic potential of the simple pendulum in a gravitational field. Fig. 9 A baryon number changing process in the presence of the emission of a large number of scalar and vector gauge quanta. [1Q Fi gure l 410 E �s EMPTY E=O ::::::} OCCUPIED GAUGE TRANSFORMATION } E=O --1--.;.--OCCUPIED LEVEL CROSSES E=O C YO U ARE HERE : ) V ��cs ::::::}EM PTY E =Q--l----+- Q OCCUPIED Figure 2 E B YOU WA NT TO Q BE HERE Figure 3 41 1 Fi gure 4 ARE HERE y /YOU x 0 ---�-- WANNA YOU. BE HERE Fi gure 5 '-..,-' c,s Figure 6 Figure 7 412 �AvIAL -2 -rr 0 2 -rr 8 Figure 8 Figure 9